The Paris prosecutor’s office said one of the men had been taken into custody as he was preparing to take a flight from Charles de Gaulle Airport.
Items worth €88m (£76m; $102m) were taken from the world’s most-visited museum last Sunday, when four thieves wielding power tools broke into the building in broad daylight.
France’s justice minister has since conceded security protocols “failed”, leaving the country with a “terrible image”.
The Paris prosecutor’s office said in a statement that the arrests had been made on Saturday evening, without specifying how many people had been taken into custody.
Specialist police can question them for up to 96 hours.
It criticised the “premature disclosure” of information related to the case, adding that it hindered efforts to recover the jewels and find the thieves.
The gang reportedly arrived at 09:30 (06:30 GMT), shortly after the museum opened to visitors.
The suspects arrived with a vehicle-mounted mechanical lift to gain access to the Galerie d’Apollon (Gallery of Apollo) via a balcony close to the River Seine.
Pictures from the scene showed the ladder leading up to a first-floor window.
Two of the thieves entered by cutting through the window with power tools.
They then threatened the guards, who evacuated the premises, and cut through the glass of two display cases containing jewels.
A preliminary report has revealed that one in three rooms in the area of the museum raided had no CCTV cameras, according to French media.
French police say the thieves were inside for four minutes and made their escape on two scooters waiting outside at 09:38.
Security measures have since been tightened around France’s cultural institutions.
